Integrand size = 10, antiderivative size = 82 \[ \int (-1+x)^{5/2} \csc ^{-1}(x) \, dx=\frac {4 x \sqrt {-1+x^2} \left (83-19 x+3 x^2\right )}{105 \sqrt {-1+x} \sqrt {x^2}}+\frac {2}{7} (-1+x)^{7/2} \csc ^{-1}(x)+\frac {4 x \text {arctanh}\left (\frac {\sqrt {-1+x^2}}{\sqrt {-1+x}}\right )}{7 \sqrt {x^2}} \]
2/7*(-1+x)^(7/2)*arccsc(x)+4/7*x*arctanh((x^2-1)^(1/2)/(-1+x)^(1/2))/(x^2) ^(1/2)+4/105*x*(3*x^2-19*x+83)*(x^2-1)^(1/2)/(-1+x)^(1/2)/(x^2)^(1/2)
Time = 0.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.88 \[ \int (-1+x)^{5/2} \csc ^{-1}(x) \, dx=\frac {4 \sqrt {1-\frac {1}{x^2}} x \left (83-19 x+3 x^2\right )}{105 \sqrt {-1+x}}+\frac {2}{7} (-1+x)^{7/2} \csc ^{-1}(x)+\frac {4}{7} \text {arctanh}\left (\frac {\sqrt {1-\frac {1}{x^2}} x}{\sqrt {-1+x}}\right ) \]
(4*Sqrt[1 - x^(-2)]*x*(83 - 19*x + 3*x^2))/(105*Sqrt[-1 + x]) + (2*(-1 + x )^(7/2)*ArcCsc[x])/7 + (4*ArcTanh[(Sqrt[1 - x^(-2)]*x)/Sqrt[-1 + x]])/7
Time = 0.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5750, 1898, 586, 25, 99, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (x-1)^{5/2} \csc ^{-1}(x) \, dx\) |
\(\Big \downarrow \) 5750 |
\(\displaystyle \frac {2}{7} \int \frac {(x-1)^{7/2}}{\sqrt {1-\frac {1}{x^2}} x^2}dx+\frac {2}{7} (x-1)^{7/2} \csc ^{-1}(x)\) |
\(\Big \downarrow \) 1898 |
\(\displaystyle \frac {2 \sqrt {x^2-1} \int \frac {(x-1)^{7/2}}{x \sqrt {x^2-1}}dx}{7 \sqrt {1-\frac {1}{x^2}} x}+\frac {2}{7} (x-1)^{7/2} \csc ^{-1}(x)\) |
\(\Big \downarrow \) 586 |
\(\displaystyle \frac {2 \sqrt {x+1} \sqrt {x-1} \int -\frac {(1-x)^3}{x \sqrt {x+1}}dx}{7 \sqrt {1-\frac {1}{x^2}} x}+\frac {2}{7} (x-1)^{7/2} \csc ^{-1}(x)\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2}{7} (x-1)^{7/2} \csc ^{-1}(x)-\frac {2 \sqrt {x-1} \sqrt {x+1} \int \frac {(1-x)^3}{x \sqrt {x+1}}dx}{7 \sqrt {1-\frac {1}{x^2}} x}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {2}{7} (x-1)^{7/2} \csc ^{-1}(x)-\frac {2 \sqrt {x-1} \sqrt {x+1} \int \left (-(x+1)^{3/2}+5 \sqrt {x+1}+\frac {1}{x \sqrt {x+1}}-\frac {7}{\sqrt {x+1}}\right )dx}{7 \sqrt {1-\frac {1}{x^2}} x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{7} (x-1)^{7/2} \csc ^{-1}(x)-\frac {2 \sqrt {x-1} \sqrt {x+1} \left (-2 \text {arctanh}\left (\sqrt {x+1}\right )-\frac {2}{5} (x+1)^{5/2}+\frac {10}{3} (x+1)^{3/2}-14 \sqrt {x+1}\right )}{7 \sqrt {1-\frac {1}{x^2}} x}\) |
(2*(-1 + x)^(7/2)*ArcCsc[x])/7 - (2*Sqrt[-1 + x]*Sqrt[1 + x]*(-14*Sqrt[1 + x] + (10*(1 + x)^(3/2))/3 - (2*(1 + x)^(5/2))/5 - 2*ArcTanh[Sqrt[1 + x]]) )/(7*Sqrt[1 - x^(-2)]*x)
3.8.2.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_) , x_Symbol] :> Simp[(a + b*x^2)^FracPart[p]/((c + d*x)^FracPart[p]*(a/c + ( b*x)/d)^FracPart[p]) Int[(e*x)^m*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0]
Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^ (q_.), x_Symbol] :> Simp[x^(2*n*FracPart[p])*((a + c/x^(2*n))^FracPart[p]/( c + a*x^(2*n))^FracPart[p]) Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + a*x^(2*n ))^p, x], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[mn2, -2*n] && !I ntegerQ[p] && !IntegerQ[q] && PosQ[n]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol ] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCsc[c*x])/(e*(m + 1))), x] + Simp[b/ (c*e*(m + 1)) Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] / ; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
Time = 0.31 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {2 \left (-1+x \right )^{\frac {7}{2}} \operatorname {arccsc}\left (x \right )}{7}+\frac {4 \sqrt {-1+x}\, \sqrt {1+x}\, \left (3 \left (-1+x \right )^{2} \sqrt {1+x}-13 \left (-1+x \right ) \sqrt {1+x}+15 \,\operatorname {arctanh}\left (\sqrt {1+x}\right )+67 \sqrt {1+x}\right )}{105 \sqrt {\frac {\left (-1+x \right ) \left (1+x \right )}{x^{2}}}\, x}\) | \(76\) |
default | \(\frac {2 \left (-1+x \right )^{\frac {7}{2}} \operatorname {arccsc}\left (x \right )}{7}+\frac {4 \sqrt {-1+x}\, \sqrt {1+x}\, \left (3 \left (-1+x \right )^{2} \sqrt {1+x}-13 \left (-1+x \right ) \sqrt {1+x}+15 \,\operatorname {arctanh}\left (\sqrt {1+x}\right )+67 \sqrt {1+x}\right )}{105 \sqrt {\frac {\left (-1+x \right ) \left (1+x \right )}{x^{2}}}\, x}\) | \(76\) |
2/7*(-1+x)^(7/2)*arccsc(x)+4/105*(-1+x)^(1/2)*(1+x)^(1/2)*(3*(-1+x)^2*(1+x )^(1/2)-13*(-1+x)*(1+x)^(1/2)+15*arctanh((1+x)^(1/2))+67*(1+x)^(1/2))/((-1 +x)*(1+x)/x^2)^(1/2)/x
Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (62) = 124\).
Time = 0.26 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.52 \[ \int (-1+x)^{5/2} \csc ^{-1}(x) \, dx=\frac {2 \, {\left (15 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )} \sqrt {x - 1} \operatorname {arccsc}\left (x\right ) + 2 \, {\left (3 \, x^{2} - 19 \, x + 83\right )} \sqrt {x^{2} - 1} \sqrt {x - 1} + 15 \, {\left (x - 1\right )} \log \left (\frac {x^{2} + \sqrt {x^{2} - 1} \sqrt {x - 1} - 1}{x^{2} - 1}\right ) - 15 \, {\left (x - 1\right )} \log \left (-\frac {x^{2} - \sqrt {x^{2} - 1} \sqrt {x - 1} - 1}{x^{2} - 1}\right )\right )}}{105 \, {\left (x - 1\right )}} \]
2/105*(15*(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)*sqrt(x - 1)*arccsc(x) + 2*(3*x^2 - 19*x + 83)*sqrt(x^2 - 1)*sqrt(x - 1) + 15*(x - 1)*log((x^2 + sqrt(x^2 - 1)*sqrt(x - 1) - 1)/(x^2 - 1)) - 15*(x - 1)*log(-(x^2 - sqrt(x^2 - 1)*sqr t(x - 1) - 1)/(x^2 - 1)))/(x - 1)
Timed out. \[ \int (-1+x)^{5/2} \csc ^{-1}(x) \, dx=\text {Timed out} \]
Time = 0.78 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.41 \[ \int (-1+x)^{5/2} \csc ^{-1}(x) \, dx=\frac {4}{35} \, {\left (x + 1\right )}^{\frac {5}{2}} - \frac {20}{21} \, {\left (x + 1\right )}^{\frac {3}{2}} + \frac {2}{7} \, {\left (x^{3} \arctan \left (1, \sqrt {x + 1} \sqrt {x - 1}\right ) - 3 \, x^{2} \arctan \left (1, \sqrt {x + 1} \sqrt {x - 1}\right ) + 3 \, x \arctan \left (1, \sqrt {x + 1} \sqrt {x - 1}\right ) - \arctan \left (1, \sqrt {x + 1} \sqrt {x - 1}\right )\right )} \sqrt {x - 1} + 4 \, \sqrt {x + 1} + \frac {2}{7} \, \log \left (\sqrt {x + 1} + 1\right ) - \frac {2}{7} \, \log \left (\sqrt {x + 1} - 1\right ) \]
4/35*(x + 1)^(5/2) - 20/21*(x + 1)^(3/2) + 2/7*(x^3*arctan2(1, sqrt(x + 1) *sqrt(x - 1)) - 3*x^2*arctan2(1, sqrt(x + 1)*sqrt(x - 1)) + 3*x*arctan2(1, sqrt(x + 1)*sqrt(x - 1)) - arctan2(1, sqrt(x + 1)*sqrt(x - 1)))*sqrt(x - 1) + 4*sqrt(x + 1) + 2/7*log(sqrt(x + 1) + 1) - 2/7*log(sqrt(x + 1) - 1)
Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (62) = 124\).
Time = 0.44 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.78 \[ \int (-1+x)^{5/2} \csc ^{-1}(x) \, dx=\frac {2}{35} \, {\left (5 \, {\left (x - 1\right )}^{\frac {7}{2}} + 21 \, {\left (x - 1\right )}^{\frac {5}{2}} + 35 \, {\left (x - 1\right )}^{\frac {3}{2}} + 35 \, \sqrt {x - 1}\right )} \arcsin \left (\frac {1}{x}\right ) - \frac {2}{5} \, {\left (3 \, {\left (x - 1\right )}^{\frac {5}{2}} + 10 \, {\left (x - 1\right )}^{\frac {3}{2}} + 15 \, \sqrt {x - 1}\right )} \arcsin \left (\frac {1}{x}\right ) + 2 \, {\left ({\left (x - 1\right )}^{\frac {3}{2}} + 3 \, \sqrt {x - 1}\right )} \arcsin \left (\frac {1}{x}\right ) - 2 \, \sqrt {x - 1} \arcsin \left (\frac {1}{x}\right ) + \frac {4 \, {\left (3 \, {\left (x + 1\right )}^{\frac {5}{2}} - 4 \, {\left (x + 1\right )}^{\frac {3}{2}} + 21 \, \sqrt {x + 1}\right )}}{105 \, \mathrm {sgn}\left ({\left (x - 1\right )}^{\frac {3}{2}} + \sqrt {x - 1}\right )} - \frac {4 \, {\left ({\left (x + 1\right )}^{\frac {3}{2}} + \sqrt {x + 1}\right )}}{5 \, \mathrm {sgn}\left ({\left (x - 1\right )}^{\frac {3}{2}} + \sqrt {x - 1}\right )} + \frac {2 \, \log \left (\sqrt {x + 1} + 1\right )}{7 \, \mathrm {sgn}\left ({\left (x - 1\right )}^{\frac {3}{2}} + \sqrt {x - 1}\right )} - \frac {2 \, \log \left (\sqrt {x + 1} - 1\right )}{7 \, \mathrm {sgn}\left ({\left (x - 1\right )}^{\frac {3}{2}} + \sqrt {x - 1}\right )} + \frac {4 \, \sqrt {x + 1}}{\mathrm {sgn}\left ({\left (x - 1\right )}^{\frac {3}{2}} + \sqrt {x - 1}\right )} \]
2/35*(5*(x - 1)^(7/2) + 21*(x - 1)^(5/2) + 35*(x - 1)^(3/2) + 35*sqrt(x - 1))*arcsin(1/x) - 2/5*(3*(x - 1)^(5/2) + 10*(x - 1)^(3/2) + 15*sqrt(x - 1) )*arcsin(1/x) + 2*((x - 1)^(3/2) + 3*sqrt(x - 1))*arcsin(1/x) - 2*sqrt(x - 1)*arcsin(1/x) + 4/105*(3*(x + 1)^(5/2) - 4*(x + 1)^(3/2) + 21*sqrt(x + 1 ))/sgn((x - 1)^(3/2) + sqrt(x - 1)) - 4/5*((x + 1)^(3/2) + sqrt(x + 1))/sg n((x - 1)^(3/2) + sqrt(x - 1)) + 2/7*log(sqrt(x + 1) + 1)/sgn((x - 1)^(3/2 ) + sqrt(x - 1)) - 2/7*log(sqrt(x + 1) - 1)/sgn((x - 1)^(3/2) + sqrt(x - 1 )) + 4*sqrt(x + 1)/sgn((x - 1)^(3/2) + sqrt(x - 1))
Timed out. \[ \int (-1+x)^{5/2} \csc ^{-1}(x) \, dx=\int \mathrm {asin}\left (\frac {1}{x}\right )\,{\left (x-1\right )}^{5/2} \,d x \]